256 HISTORY OF THE THEORY OF NUMBERS. [CHAP, vm
according as the upper or lower sign holds. If Z=3m+l, Q= — 1, then
IfZ=3m— 1,0=— 1, there are three real roots if and only if a/6 is a cubic residue of I, viz., 2/w=0; when real, the roots may be found as in the second
Cailler142 noted that a cubic_equation X = 0 has its roots expressible
rationally in one root and A/A, where A is the discriminant (Serret's Alg&bre, ed. 5, vol. 2, 466-8). Hence, if p is a prime, JSTssO (mod p) has three real roots if one, when and only when A is a quadratic residue of p. If pssQm*!, his141 test shows that x3— 3#+l=0 (mod p) has three real roots, but no real root for other prime moduli 5*3. The function F(xj =x34-£2— 2z— 1 for the three periods of the seventh roots of unity is divisible by the prunes 7m=± 1 (then 3 real roots, Gauss60, p. 624) and 7, but by no other primes.
E. B. Escott143 noted that the equation F(x) = 0 last mentioned has the roots a, )3 = a2— 2, 7=/32— 2, so that F(x)=0 (mod p) has three real roots if one real root. To find the most general irreducible cubic equation with roots a, ft 7 such that
we may assume that /(x) is of degree 2. For /(a) = a2— n, we get (2) x3+ax2-(a2-2a+3)a;-(a3-2a2+3a~l)=0,
with /3 = a2— c, 7=/32— c, a=72—c, c=a2 — a +2. The corresponding congruence has three real roots if one. To treat/(a)=a2-f&a+Z, add k/2 to each root. For the new roots, ft' = a'2— n, as in the former case. To treat /(a) = ta2+ga+h, the products of the roots by t satisfy the preceding relation. L. E. Dickson144 determined the values of a for which the congruence corresponding to (2) has three integral roots. Replace x by z— a; we get
^-2a^+(2a-3)z+l==0 (mod p).
If one root is z, the others are 1 — 1/2 and 1/(1 —z). Evidently a is rational in z. If —3 is a quadratic non-residue of p} there are exactly (p — 2)/3 values of a for which the congruence has three distinct integral roots. If —3 is a residue, the number is (p+2)/3. A second method, yielding an explicit congruence for these values of a, is a direct application of his138 general criteria for the nature of the roots of a cubic congruence.
T. Hayashi145 treated cyclotomic cubic equations with three real roots by use of EscottV43 results.
l42L'intenn£diaire des math., 16, 1909, 185-7. MIbid., (2), 12, 1910-11, 149-152.
"•Annals of Math., (2), 11, 1909-10, 85-92. ™Ibid.t 189-192.